When he first hears of the Ring and why it can't stay in the Shire, Frodo only asks, “What must I do?” With very little hesitation, he decides to take the Ring to Rivendell, where the Elves will decide what to do with it. Once there, he is intrusted with a very challenging quest – to take the Ring across Middle-Earth and destroy it. It sounds far from easy – in fact, he knows he'll probably die in the process – but he decides to take the challenge with a simple, “I will take the Ring to Mordor, though I do not know the way.” The journey is certain to get dark, lonely, and depressing, and he needs someone to encourage him, especially now, at his darkest hour.
This is where Sam comes in.
Frodo and Sam's relationship is meaningful because it's one of the key things that keeps the plot of Lord of the Rings moving forward. Yes, a lot of the book is about the return of the king and the rising up of minorities – such as the Ents and hobbits – who join together to conquer evil and restore peace. Very little of this could happen, however, if it weren't for Frodo and Sam and their mission (...quest...thing) to destroy the Ring. Frodo keeps pushing onward to complete his goal of dropping the Ring into the lava in Mount Doom, and Sam encourages Frodo the whole way.
Their friendship is relevant in today's society because it forms a model of what a good friendship looks like, especially on Sam's side. After all, it was Sam who said those inspiring words at the end of The Two Towers:
Frodo and Sam's relationship inspires us to seek this kind of relationship today – in our families, friends, and even our enemies, as Sam did in the movie after Frodo tells him to go home. Their friendship is forever formed and they stay friends long after they complete their mission, something we, too, should strive for.